On convergence to the Denjoy-Wolff point

Abstract

For holomorphic selfmaps of the open unit disc $\U$ that are not elliptic automorphisms, the Schwarz Lemma and the Denjoy-Wolff Theorem combine to yield a remarkable result: each such map $\phi$ has a (necessarily unique) ``Denjoy-Wolff point'' $\dwp$ in the closed unit disc that attracts every orbit in the sense that the iterate sequence $(\phin)$ converges to $\dwp$ uniformly on compact subsets of $\U$. In this paper we prove that, except for the obvious counterexamples—inner functions having $\dwp\in\U$—the iterate sequence exhibits an even stronger affinity for the Denjoy-Wolff point; $\phin\goesto\dwp$ in the norm of the Hardy space $H^p$ for $1\le p<\infty$. For each such map, some subsequence of iterates converges to $\dwp$ almost everywhere on $\bdu$, and this leads us to investigate the question of almost-everywhere convergence of the entire iterate sequence. Here our work makes natural connections with two important aspects of the study of holomorphic selfmaps of the unit disc: linear-fractional models and ergodic properties of inner functions.